Upper-bounding the k-colorability threshold by counting covers

TL;DR

This paper rigorously establishes an upper bound on the average degree for the non-$k$-colorability of random graphs, confirming a physics-inspired conjecture through the first moment method applied to graph covers.

Contribution

It provides the first rigorous proof matching the conjectured threshold for non-$k$-colorability in random graphs using the concept of covers.

Findings

Proves non-$k$-colorability for $d > 2k \ln k - \\ln k - 1 + o_k(1)$ with high probability.

Matches the conjectured threshold from statistical physics predictions.

Introduces a proof technique based on counting covers, a physics-inspired concept.

Abstract

Let $G(n,m)$ be the random graph on $n$ vertices with $m$ edges. Let $d=2m/n$ be its average degree. We prove that $G(n,m)$ fails to be $k$-colorable with high probability if $d>2k\ln k-\ln k-1+o_k(1)$. This matches a conjecture put forward on the basis of sophisticated but non-rigorous statistical physics ideas (Krzakala, Pagnani, Weigt 2004). The proof is based on applying the first moment method to the number of "covers", a physics-inspired concept. By comparison, a standard first moment over the number of $k$-colorings shows that $\gnm$ is not $k$-colorable with high probability if $d>2k\ln k-\ln k$.